But we're talking about air-breathing rockets, not merely engines. Crucially, a rocket supplies its own oxidiser, and an air-breathing rocket supplements this with intake air. An example of such would be Sabre: https://en.wikipedia.org/wiki/SABRE_%28rocket_engine%29

Ok, you are reacting to the exact term the GP used, but it doesn't look like it was used as a term of trade, just as a general characteristic.

Using an airplane as a first stage of a rocket gives the exact same air-breathing advantages as using a rocket (or better, because the planes uses air as reaction mass too). That makes both equivalent on practice. Yes, an hypersonic engine makes for a larger stage, and an dual one that can operate with and without air makes for an even larger stage, but they are of the same broad category.